In this paper we introduce the Riemannian Convex Potential Map (RCPM),
a generic model for generative modeling on arbitrary
Riemannian manifolds that enjoys universal representational power.
RCPM (illustrated in fig:demo) is based on Optimal Transport (OT) over Riemannian
manifolds [1]}, [2]}, [3]}, [4]}
and generalizes the convex potential flows in the Euclidean
setting by [5]}.
We prove that RCPMs are universal on any compact
Riemannian manifold, which comes from the fact that our discrete
\(c\) -concave potential functions are universal.
Our experimental demonstrations show that RCPMs are competitive
and model standard distributions on spheres and tori.
We further show a case study in modeling continental drift where
we transport Earth's land mass on the sphere.
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